College Algebra, 2013a

3.1 Graphs of Polynomials 241
Despite having different end behavior, all functions of the form f(x) =ax n for natural numbers n
share two properties which help distinguish them from other animals in the algebra zoo: they are
continuous and smooth. While these concepts are formally defined using Calculus, 14 informally,
graphs of continuous functions have no ‘breaks’ or ‘holes’ in them, and the graphs of smooth
functions have no ‘sharp turns’. It turns out that these traits are preserved when functions are
added together, so general polynomial functions inherit these qualities. Below we find the graph of a
function which is neither smooth nor continuous, and to its right we have a graph of a polynomial,
for comparison. The function whose graph appears on the left fails to be continuous where it
has a ‘break’ or ‘hole’ in the graph; everywhere else, the function is continuous. The function is
continuous at the ‘corner’ and the ‘cusp’, but we consider these ‘sharp turns’, so these are places
where the function fails to be smooth. Apart from these four places, the function is smooth and
continuous. Polynomial functions are smooth and continuous everywhere, as exhibited in the graph
on the right.
‘corner’
‘cusp’
‘hole’
‘break’
Pathologies not found on graphs of polynomials
The graph of a polynomial
The notion of smoothness is what tells us graphically that, for example, f(x) =|x|, whose graph is
the characteristic ‘∨’ shape, cannot be a polynomial. The notion of continuity is what allowed us
to construct the sign diagram for quadratic inequalities as we did in Section 2.4. This last result is
formalized in the following theorem.
Theorem 3.1. The Intermediate Value Theorem (Zero Version): Suppose f is a continuous
function on an interval containing x = a and x = b with a